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Beliebt Algebra >

x(x^3+i)=i(x-1)+sqrt(3)

Lösung

x (x^{3} + i) = i (x - 1) + 3

Lösung

x = \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4} - i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4}, x = \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4} + i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4}, x = - \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4} + i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4}, x = - \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4} - i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4}

Schritte zur Lösung

x (x^{3} + i) = i (x - 1) + 3

Schreibe

x (x^{3} + i)

um:

x^{4} + i x

x^{4} + i x = i (x - 1) + 3

Verschiebe

i (x - 1)

auf die linke Seite

x^{4} + i = 3

Verschiebe

i

auf die rechte Seite

x^{4} = 3 - i

Für

z^{n} = a

sind die Lösungen

z_{k} = n ∣ a ∣ (cos (\frac{ar g ( a ) + 2 kπ}{n}) + i sin (\frac{ar g ( a ) + 2 kπ}{n})),

k = 0, 1, \dots, n - 1

Für

n = 4, a = 3 - i

∣ a ∣ = 2

ar g (a) = - \frac{π}{6}

x = 42 (cos (\frac{- \frac{π}{6} + 2 \cdot 0 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 0 π}{4})), x = 42 (cos (\frac{- \frac{π}{6} + 2 \cdot 1 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 1 π}{4})), x = 42 (cos (\frac{- \frac{π}{6} + 2 \cdot 2 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 2 π}{4})), x = 42 (cos (\frac{- \frac{π}{6} + 2 \cdot 3 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 3 π}{4}))

Vereinfache

42 (cos (\frac{- \frac{π}{6} + 2 \cdot 0 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 0 π}{4})) : \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4} - i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4}

Vereinfache

42 (cos (\frac{- \frac{π}{6} + 2 \cdot 1 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 1 π}{4})) : \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4} + i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4}

Vereinfache

42 (cos (\frac{- \frac{π}{6} + 2 \cdot 2 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 2 π}{4})) : - \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4} + i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4}

Vereinfache

42 (cos (\frac{- \frac{π}{6} + 2 \cdot 3 π}{4}) + i sin (\frac{- \frac{π}{6} + 2 \cdot 3 π}{4})) : - \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4} - i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4}

x = \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4} - i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4}, x = \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4} + i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4}, x = - \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4} + i \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4}, x = - \frac{2 ^{\frac{3}{4}} 4 - 2 - 6}{4} - i \frac{2 ^{\frac{3}{4}} 4 + 2 + 6}{4}

Beliebte Beispiele

x^{3} + 6 x = 0

16m^4+48m^2+36=0

16 m^{4} + 48 m^{2} + 36 = 0

9 - 16 x^{5} - 5 x = 7

(x^2-7x+11)x^2-13x+42=1

(x^{2} - 7 x + 11) x^{2} - 13 x + 42 = 1

x^{(4)}-13x^{(2)}+36=0

x^{(4)} - 13 x^{(2)} + 36 = 0